C^\infty smoothing for weak solutions of the inhomogeneous Landau equation

We consider the spatially inhomogeneous Landau equation with initial data that is bounded by a Gaussian in the velocity variable. In the case of moderately soft potentials, we show that weak solutions immediately become smooth and remain smooth as long as the mass, energy, and entropy densities remain under control. For very soft potentials, we obtain the same conclusion with the additional assumption that a sufficiently high moment of the solution in the velocity variable remains bounded. Our proof relies on the iteration of local Schauder-type estimates.Comment: 23 pages, updated with to-be-published versio

Understanding long-term English language learners in the developmental classroom of a community college

This qualitative, phenomenological study is focused on 13 identified LTELLs currently enrolled in a community college classroom near the Texas-Mexico border. Both male and female students were identified and invited for participation based on an initial questionnaire Through an emic, qualitative lens, this research explores the beliefs that LTELLs hold with regard to language, educational supports, and gender in college. Second, the research looks to understand how these beliefs may impact the education of LTELLs currently enrolled in developmental classes of a community college. The driving overarching questions were: (1.) How do LTELLs perceive their own use of language? (2.) How do LTELLs identify and understand their social and cultural supports in attending community college? (3.) How do male LTELLs perceive gender differences in the classroom? How do female LTELLs perceive gender differences in the classroom? In order to answer these questions as thoroughly as possible, the research was designed to gain insight directly from students through interviews. Through purposeful sampling of 63 students enrolled in four sections of Integrated Reading and Writing during the spring semester of 2014, a pool of potential participants was identified. The pool was limited to the initial identification of LTELLs based on established characteristics and their answers to the distributed questionnaires. The student-participants were interviewed two times over the course of the 16 week semester: beginning and end. The initial semi-structured interview primarily explored students\u27 educational experiences. The final semi-structured exit-interview asked for students\u27 educational experiences and reflections from the semester and gives students a chance to discuss the things that enabled or hindered their success during their semester or semesters at college. Findings from the research focused on three areas: language, educational supports, and gender. For supports, the LTELLs in this study recognized the importance of family as a pushing force in their educational attainment, as well as the cultural capital of extended family. LTELLs also showed a more realistic self-assessment in terms of their academic abilities. Finally, gender discrepancies between males and females were perceived as the result of underlying motivations. (PsycInfo Database Record (c) 2022 APA, all rights reserved

Existence and stability of near-constant solutions of variable-coefficient scalar field equations

This article studies a class of semilinear scalar field equations on the real line with variable coefficients in the linear terms. These coefficients are not necessarily small perturbations of a constant. We prove that under suitable conditions, the non-translation-invariant linear operator leads to steady states that are ``almost constant'' in the spatial variable. The main challenge of the proof is due to a spectral obstruction that cannot be treated perturbatively. Next, we consider stability of constant and near-constant steady states. We establish asymptotic stability for the vacuum state with respect to perturbations in $H^1\times L^2$, without placing any parity assumptions on the coefficients, potential, or initial data. Finally, under a parity assumption, we show asymptotic stability for near-constant steady states.Comment: 25 pages, final accepted version. Previous title: "Asymptotic stability for near-constant solutions of variable-coefficient scalar field equations.